[Paper Review] Exact Solutions for M/M/c/Setup Queues

Tuan Phung-Duc|arXiv (Cornell University)|Jun 12, 2014

Advanced Queuing Theory Analysis24 references4 citations

TL;DR

This paper presents exact analytical solutions for the joint stationary queue length distribution in M/M/c/Setup queues with ON-OFF server activation policy using two methodologies: generating function and matrix analytic methods. It derives closed-form expressions for generating functions, factorial moments, and conditional decomposition, while also providing an efficient recursive algorithm with O(c²) complexity, offering deep insights into power-performance trade-offs in data centers.

ABSTRACT

Recently multiserver queues with setup times have been extensively studied because they have applications in power-saving data centers. The most challenging model is the M/M/$c$/Setup queue where a server is turned off when it is idle and is turned on if there are some waiting jobs. Recently, Gandhi et al.~(SIGMETRICS 2013, QUESTA 2014) present the recursive renewal reward approach as a new mathematical tool to analyze the model. In this paper, we derive exact solutions for the same model using two alternative methodologies: generating function approach and matrix analytic method. The former yields several theoretical insights into the systems while the latter provides an exact recursive algorithm to calculate the joint stationary distribution and then some performance measures so as to give new application insights.

Motivation & Objective

• To derive exact closed-form solutions for the joint stationary queue length distribution in M/M/c/Setup queues with ON-OFF policy.
• To compare and contrast two analytical methodologies—generating function and matrix analytic methods—for solving the model.
• To provide performance insights into power consumption, switching rates, and system behavior under varying traffic and setup parameters.
• To establish the equivalence and computational advantages of both approaches over prior recursive renewal reward methods.

Proposed method

• Employing the generating function approach to derive exact expressions for the joint stationary distribution, generating functions, and factorial moments of any order.
• Using the matrix analytic method to construct an explicit recursive algorithm based on the rate matrix (R) and first passage probability matrix (G), enabling efficient computation.
• Exploiting the special structure of the non-homogeneous part of the underlying Markov chain to reduce computational complexity to O(c²) for the generating function method and O(c³) for the matrix analytic method.
• Applying conditional decomposition to reveal structural properties of the queue length distribution.
• Validating the equivalence of both methodologies through analytical and numerical comparisons.
• Adapting the framework to variant models such as threshold-based policies and fixed-on server configurations.

Experimental results

Research questions

• RQ1 How can the joint stationary queue length distribution be exactly solved for the M/M/c/Setup queue with ON-OFF policy using generating functions?
• RQ2 What is the computational complexity and numerical efficiency of the matrix analytic method compared to prior recursive renewal reward approaches?
• RQ3 How do power consumption and switching rates vary with traffic intensity ρ and setup rate α in the ON-OFF policy?
• RQ4 Under what parameter regimes does the ON-OFF policy outperform the ON-IDLE policy in terms of total cost?
• RQ5 Can the proposed methodologies be extended to other variants, such as threshold-based or fixed-on server policies?

Key findings

• The generating function approach yields exact closed-form expressions for the joint stationary queue length distribution, generating functions, and factorial moments of any order.
• The matrix analytic method provides an efficient recursive algorithm with O(c³) complexity, enabling accurate computation of performance measures.
• The ON-OFF policy outperforms the ON-IDLE policy in terms of total cost when the setup cost ratio rs = Cs/Ca is below a critical threshold rρ, which decreases with increasing traffic intensity ρ.
• Switching rate E[Sr] increases with ρ in light traffic but decreases in heavy traffic, indicating optimal performance in extreme traffic regimes.
• For high setup rates (α = 10, 100), total power consumption initially increases with ρ but then decreases due to reduced server activation frequency in heavy loads.
• The model belongs to a special QBD class where the rate matrix R and boundary structure allow significant complexity reduction compared to general methods.